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What Is a Tree Diagram? (Mathematics Lesson)

What Is a Tree Diagram?

A tree diagram displays the possible outcomes from an event, and allows the probabilities of different outcomes to be calculated.

A Real Example of a Tree Diagram

The tree diagram below represents the tossing of a coin:



  • There are two outcomes of the coin toss: Heads and Tails.

  • There is a different branch for each outcome.

  • The probability of each outcome is written by the branch.

  • The probabilities of each branch add up to 1.
Read more about how to draw a tree diagram

Another Real Example of a Tree Diagram

Tree diagrams are useful for showing muliple events. The tree diagram below shows the possible outcomes of two tosses of a coin:



The four possible outcomes of the two coin tosses are:

  • Head, Head

  • Head, Tail

  • Tail, Head

  • Tail, Tail
It is then possible to calculate probabilities.

How to Find Probabilities of Each Outcome from a Tree Diagram

To find the probability of each outcome ('Head, Head', 'Head, Tail' etc. ) multiply along the branches.

Question: What is the probability of getting a Head in the 1st coin toss, and a Head in the 2nd coin toss?

Multiply the probabilities along the branches:





The probability of getting Heads, Heads is ¼. By a similar process, it can be shown that the probability of each outcome is ¼.



How to Find Probability of Several Outcomes from a Tree Diagram

By multiplying along each branch, we find the probability of each outcome.

But we may be interested in finding the probability that one of several outcomes may occur. For example, we might be interested in the probability of the same side landing face up in both tosses. In that case, there are two possible outcomes:

  • Head, Head

  • Tail, Tail
In this case, add the probabilities of the two outcomes.

Question: What is the probability that the coin lands the same way on both tosses (i.e. either 'Head, Head' or 'Tail, Tail')?

Add the probabilities of the two outcomes.





The probability of the coin coming up the same way twice is ½.

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Note
WHAT IS PROBABILITY?

Probability tells us how likely something is to happen.

The probability of an event is defined as:



Probability is given as a number between 0 and 1.

A probability of 0 means an event is impossible.

A probability of 1 means an event is certain.

The closer a probability is to 0 the less likely it is. The closer a probability is to 1 the more likely it is.



SIZE OF TREE DIAGRAMS

Tree diagrams can be any size.

It would be possible to draw a tree diagram for three coin tosses:



The same process applies. To find the probability of each outcome (i.e 'HHH', 'HHT' etc), multiply along the branches.

Then add across the outcomes. For instance to find the probability of 'HHH' or 'HHT', add the probability of each outcome.

Top Tip

A USEFUL CHECK

There is a useful way to check that the probabilities on a tree diagram are all right.

Firstly, notice that at each branch, the probabilities add up to 1:



Read more about adding fractions

Secondly, when the probabilities of each final outcome are known, they also add up to 1:



Read more about multiplying fractions

MULTIPLY ALONG BRANCHES, ADD ACROSS THEM

It can be confusing when to multiply probabilities and when to add them. The general rule is:

  • Multiply along the branches. This gives the overall probability of each outcome.

  • Add across the branches. This gives the probability of that this outcome or that outcome would occur.